Matthew Schwartz Statistical Mechanics, Spring 2019 Lecture 13: Metals 1 Introduction Fermions satisfy the Pauli-exclusion principle: no two fermions can occupy the same state. This makes fermionic systems act very dierently from bosonic systems. We introduced the non-interacting Fermi gas in Lecture 10, and will continue to discuss it here. We assume the fermions do not interact with each other. This means that the single-particle states of a system are not aected by which states are occupied. So we will x the states, then ll them from the bottom up, then see what happens when we heat it up. You can think of the possible states as bunch of little shelves and we stack the fermions onto them one-by-one. The exclusion principle implies that once a state is occupied, it cannot be occupied again: the occupancy number of each state is either 0 or 1. This leads to the single-particle grand partition function of state i ("i )n ("i ) = e¡ ¡ = 1 + e¡ ¡ (1) Zi n=0;1 X and grand free energy for state i: 1 1 (" ) = ln = ln[1 + e¡ i¡ ] (2) i ¡ Zi ¡ The occupation number for state i we then found to be given by (3) @i 1 ni = = f("i) = ("i ) h i ¡ @ e ¡ + 1 This is known as the Fermi-Dirac distribution or Fermi function, f("). As T 0 the function ! approaches a step function where all the states below are occupied and the staes above are unoccupied. The value of at T = 0 is called the Fermi energy and denoted by "F . Sometimes the Fermi energy is also called the Fermi level. At any temperature T > 0, some fermions will have energies higher than "F , and therefore some states with energy less than "F will not be occupied. We call the unoccupied states holes. You can picture a nite-temperature Fermi gas as having fermions constantly bouncing out of lower-energy states into higher energy states, leaving holes, then other fermions falling out of excited states into those holes. The electron/hole picture gives a useful way to think about nite temperature metals and gives a powerful way to understand the physics of semiconductors, as we will see in this and the next lecture. We call the temperature corresponding to the Fermi energy the Fermi temperature: "F TF (4) kB 1 2 Section 2 When the temperature of a system is well below the Fermi temperature, T T , then " and F F the Fermi distribution looks very much like a step function f(") (" "). In such situations, the F ¡ lowest states are almost all occupied and we say the system is highly degenerate. In a degenerate system, quantum eects are important. Generally, Fermi temperatures in metals are very high (for example, TF for electrons in copper is around 80,000K, as we will see below). So high in fact, that it is generally impossible to heat a system above the Fermi temperature without changing the nature of the system (i.e. melting the solid). Thus, quantum statistics are always important for electrons in metals. The typical picture of a highly degenerate fermionic system is having a relatively small set of energy levels that are relevant, namely those near " . Energy levels with " " require a lot of F F energy, of order "F , to be excited into an empty state, since the rst available state is above "F . This is nearly impossible when T T . The set of low-energy states that do not participate in the F thermal activity are called the Fermi sea. The Fermi sea can be very deep (a lot of states), but its depth is largely irrelevant to the thermal properties of the material. Similarly, the levels with energies much higher than " are impossible to excite when T T . So only those levels near " F F F matter. Thus degenerate gases have something very close to a symmetry between states above and below "F : a symmetry between electrons and holes. We'll use this picture to understand some properties of metals. 2 Free electron gas The possible single-particle states of the electrons in a free electron gas are the same as for a boson in a box. The allowed wavenumbers are ~k = ~n; ~n = triplet of whole numbers (5) L In the non-relativistic limit energies are determined by the usual relation in quantum mechanics 2~ 2 2 2 ~ k ~ 2 "n = = 2 n (6) 2me 2me L with n = ~n . So j j L 2m L 2m d" n = e p"; dn = e (7) ~2 2 ~2 p" r r For electrons, there are two spins. So we compute the density of states via 1 V 2m 3/2 2 2 4n2dn = e p"d" (8) ! 8 22 2 n ~ X Z Z 1 As before, the 8 accounting for the modes only counting the rst octant (i.e. the ~n are whole numbers) while the integral over the sphere includes all 8 octants. We'll also study the relativistic and ultrarelativistic limits when we discuss white dwarf stars in Lecture 15, but in metals the 3/2 electrons are generally non-relativistic, as we will check shortly. Thus g(") = V 2me p". This 22 ~2 scaling g(") p" is the same as for the Bose gas from last lecture, only the constant in g(") is ¡ dierent. 1; " < "F Recall that as T 0, the Fermi function becomes a step function: n" . The Fermi ! h i ! 0; " > "F energy "F is the chemical potential at T = 0 or equivalently the energy below which all the states are occupied at T = 0. So in particular, the number of of states at T = 0 is 3/2 "F 3/2 1 T =0 V 2me V 2me 2 3/2 N = n" g(")d" 2 2 p"d" = 2 2 "F (9) 0 h i !!!!!!!!!!!!! 2 ~ 0 2 ~ 3 Z Z Sommerfeld free electron model for metals 3 Therefore 2 N 2/3 " = ~ 32 (10) F 2m V e Using this, it's convenient to write the density of states as V 2m 3/2 3N g(") = e p" = p" (11) 22 ~2 3/2 2"F The energy of the Fermi gas at zero temperature is 3N "F 3 E = E(T = 0) = "p"d" = N" (12) 0 3/2 5 F 2"F Z0 E 3 So the average energy of each electron is N = 5 "F . An important concept in electron gases, or Fermi gases in general, is that of degeneracy pressure. Pressure is dened as P = @E . Normally we associate pressure with the kinetic motion ¡ @V of molecules bombarding the walls of a container. This exerts a force on the walls, so that if the volume increases, the energy would go down. Because the force is due to thermal motion, this ordinary pressure is absent at T = 0. Degeneracy pressure, on the other hand, is a contribution to @E in Fermi gases that persists even at T = 0. ¡ @V The way to understand degeneracy pressure is by thinking about how the energy levels change as the volume is shrunk, for example under isothermal compression. For a classical gas, isothermal compression does not change the energy of the gas (i.e. E = CVkBT is volume independent). What it does is increase the density of molecules, so the number of collisions on the wall of a container N goes up, and hence the pressure increases (P = V kBT ). For a Bose gas, as the volume shrinks, all the energy levels go up, but the occupancies of dierent levels adjust so that the total energy is the same. Thus, Bose gases behave essentially like classical gases when compressed. For a degenerate Fermi gas, when the volume goes down, the occupancies of states cannot adjust. Instead, the energy levels go up and the total energy simply increases. This contribution to @E is called degeneracy ¡ @V pressure. To see that this pressure really is dierent from kinetic motion on container walls, consider the T = 0 limit. As we saw before, T = 0 is often a good approximation when T T . Since we have F been working at T = 0 already, all we have to do is dierentiate Eq. (12): @E @ 3 3 @ 2 N 2/3 2 E P = = N" = N ~ 32 = 0 (13) ¡ @V ¡@V 5 F ¡5 @V 2m V 3 V " e # In this limit, the classical pressure goes to zero but the degeneracy pressure persists. Degeneracy pressure is essential to understand white dwarfs and neutron stars, as we'll see in Lecture 15. 3 Sommerfeld free electron model for metals Sommerfeld proposed that the electrons in a metal can be described by a free electron gas. The basic idea is that each atom in a metal has some number of loosely bound electrons that it contributes to the gas. The alkali metals (rst column of periodic table, e.g. potassium) are monovalent, contributing one electron each. Some metals contribute more than one electron per atom (we'll see why in Lecture 15). The weakly bound electrons, called the valence electrons, are assumed to be free at leading order, so they just bounce around like particles in a box, with the size of the box determined by the size of the metal. This is called the free electron model, 4 Section 3 Before getting into the predictions of the free electron model, it's worth a few comments about why on earth this model should be at all reasonable. At rst thought, it seems crazy that we could both ignore the attractive interactions of the electrons to the positively charged nuclei and ignore the repulsive Coulomb interactions among the electrons themselves.